HEY, what is the name for H3C?

is it negatively charged methane?

Do you mean methyl group? Methane is CH₄.

That would be a positively charged methyl carbocation. CH3+

That would be a positively charged methyl carbocation. CH3+

 

It could just as easily be the carbanion.

It could just as easily be the carbanion.

 

Though as written (without charge) it is a methyl radical .

When it's attached to an organic molecule, it's the methyl group. It would be written like that (H3C) because it's on the left side, and usually the organic molecules are shown with hydrogens towards the outside and the carbon closest to the bond.